insurance-monitoring 0.9.4

Model drift detection, calibration monitoring, and sequential A/B testing for insurance pricing models: PSI, Gini drift, Murphy decomposition, mSPRT champion/challenger.

insurance-monitoring

Models drift. Regulators notice.

Blog post: Insurance Model Monitoring: Beyond Generic Drift Detection

A pricing model that is 15% cheap on young drivers and 15% expensive on mature drivers reads 1.00 at portfolio level — and triggers no alarm. The loss ratio deteriorates twelve months later. By then, the PRA expects regulated insurers to have model risk management frameworks (and SS1/23, while primarily aimed at banks, is widely adopted as the de facto standard for insurers too) — and the framework should have caught this first. insurance-monitoring detects per-feature distribution shifts and model discrimination drift, not just the headline A/E ratio, so you find the problem before it reaches the accounts or the supervisor.

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Part of the Burning Cost stack

Takes the outputs of any fitted pricing model and a stream of actual experience. Feeds drift signals and calibration verdicts into insurance-governance for model risk committee packs. Pairs with insurance-fairness to monitor per-protected-group A/E ratios under Consumer Duty. → See the full stack

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Manual monitoring vs insurance-monitoring

Task	Manual approach	insurance-monitoring
Population Stability Index	Excel macro per factor, re-coded each quarter, unweighted	psi() / csi() — exposure-weighted, Polars-native, traffic-light band
Feature drift heatmap	Engineer writes one-off script; no standard thresholds	csi() — one call, all rating factors, PRA-aligned thresholds
A/E ratio with CI	Custom formula in SQL, no confidence interval	ae_ratio_ci() — Wilson CI, exposure-weighted, RAG status
Discrimination drift	Gini computed ad hoc; no test for statistical significance	gini_drift_test() / GiniDriftBootstrapTest — arXiv 2510.04556 bootstrap CI, governance plot
RECALIBRATE vs REFIT decision	Actuary judgment call, not documented	MonitoringReport.recommendation — arXiv 2510.04556 decision tree, Murphy decomposition sharpens it
Repeated monthly testing inflation	H-L / A/E tested afresh each month; inflated false-positive rate	PITMonitor — anytime-valid, P(ever false alarm
Champion/challenger A/B test	Wait for pre-specified sample size; cannot stop early	SequentialTest — mSPRT, valid at every interim check, supports frequency/severity/loss ratio
Drift attribution (which feature explains the performance change?)	PSI-by-eye; no interaction-aware method	DriftAttributor / InterpretableDriftDetector — TRIPODD, FDR control, exposure weighting
Monitoring report for model risk committee	Manual Word document	MonitoringReport.to_polars() — flat DataFrame, writes directly to Delta table

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Quick start

import numpy as np
import polars as pl
from insurance_monitoring import MonitoringReport
from insurance_monitoring.drift import psi

rng = np.random.default_rng(42)
# Training period
actual_ref = rng.poisson(0.08, 10_000).astype(float)
predicted_ref = np.full(10_000, 0.08)
exposure_ref = rng.uniform(0.5, 1.0, 10_000)
# Current period — model has drifted
actual_cur = rng.poisson(0.11, 5_000).astype(float)   # loss rate up
predicted_cur = np.full(5_000, 0.08)                   # model unchanged
exposure_cur = rng.uniform(0.5, 1.0, 5_000)

report = MonitoringReport(
    reference_actual=actual_ref, reference_predicted=predicted_ref,
    current_actual=actual_cur, current_predicted=predicted_cur,
    exposure=exposure_cur, murphy_distribution="poisson",
)
print(report.recommendation)          # => 'RECALIBRATE' or 'REFIT'
print(report.to_polars())             # flat DataFrame: metric / value / band

See examples/quickstart.py for a fully self-contained example with feature drift and CSI.

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Installation

pip install insurance-monitoring
# or
uv add insurance-monitoring

Dependencies: polars, numpy, scipy, matplotlib

MLflow integration (optional):

pip install insurance-monitoring mlflow

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Features

• PSI / CSI — Population Stability Index and Characteristic Stability Index across all rating factors; exposure-weighted; traffic-light bands aligned with PRA SS1/23 model risk expectations
• A/E ratio with confidence interval — Wilson CI, exposure-weighted, RAG status; the primary calibration metric for UK pricing teams
• Gini drift test — two-sample and one-sample bootstrap designs (arXiv 2510.04556); percentile CI; governance plot suitable for model validation packs
• Decision tree (MonitoringReport.recommendation) — NO_ACTION / MONITOR_CLOSELY / RECALIBRATE / REFIT / INVESTIGATE; Murphy decomposition sharpens the RECALIBRATE vs REFIT distinction
• Murphy decomposition — decomposes scoring loss into uncertainty, discrimination (DSC), and miscalibration (MCB); when DSC falls, REFIT; when MCB dominates, RECALIBRATE
• Anytime-valid calibration monitoring (PITMonitor) — probability integral transform e-processes (Henzi, Murph, Ziegel 2025); valid type I error control at every monthly check, forever; solves repeated-testing inflation
• Sequential A/B testing (SequentialTest) — mixture SPRT (Johari et al. 2022); supports Poisson frequency, log-normal severity, and compound loss ratio; no pre-specified sample size
• TRIPODD drift attribution (DriftAttributor, InterpretableDriftDetector) — feature-interaction-aware; identifies which factors explain model performance degradation; Bonferroni or Benjamini-Hochberg FDR control; exposure weighting; Poisson deviance loss
• MLflow integration (MonitoringTracker) — logs all metrics and bands to an MLflow run; requires mlflow (optional dependency)
• Polars-native throughout — no pandas required; flat to_polars() output writes directly to Delta tables on Databricks

Modules

MonitoringReport

The single entry point for a complete monthly or quarterly monitoring run.

from insurance_monitoring import MonitoringReport

report = MonitoringReport(
    reference_actual=train_claims,
    reference_predicted=train_predicted,
    current_actual=current_claims,
    current_predicted=current_predicted,
    exposure=current_exposure,
    feature_df_reference=train_features,    # pl.DataFrame
    feature_df_current=current_features,    # pl.DataFrame
    features=["driver_age", "vehicle_age", "ncd_years"],
    murphy_distribution="poisson",          # Murphy decomposition — always available
    gini_bootstrap=True,                    # percentile CI on Gini (v0.6.0)
)
print(report.recommendation)   # 'NO_ACTION' | 'RECALIBRATE' | 'REFIT' | 'INVESTIGATE'
print(report.to_dict())        # nested dict — JSON serialisable, log to MLflow
print(report.to_polars())      # flat DataFrame — write to Delta table

Recommendation logic follows the three-stage decision tree from arXiv 2510.04556:

• Gini OK + A/E OK → NO_ACTION
• A/E bad only → RECALIBRATE (update intercept)
• Gini bad → REFIT (rebuild on recent data)
• Murphy decomposition present: overrides the heuristic when miscalibration vs discrimination are unambiguous

drift — PSI, CSI, KS, Wasserstein

from insurance_monitoring.drift import psi, csi, ks_test, wasserstein_distance
import polars as pl

# PSI on a single feature — exposure-weighted
drift_val = psi(
    reference=driver_age_train,
    current=driver_age_now,
    n_bins=10,
    exposure_weights=exposure_now,    # car-years, not policy count
    reference_exposure=exposure_train,
)
print(f"PSI: {drift_val:.3f}")   # < 0.10 green | 0.10–0.25 amber | > 0.25 red

# CSI heatmap across all rating factors — returns pl.DataFrame with band column
csi_df = csi(
    reference_df=train_df,
    current_df=current_df,
    features=["driver_age", "vehicle_age", "ncd_years", "vehicle_group"],
)
print(csi_df)  # feature | csi | band

Use PSI/CSI for operational dashboards. Use ks_test for formal hypothesis testing at quarter-end (note: over-sensitive at n > 500k). Use wasserstein_distance when communicating to non-technical stakeholders — it reports drift in original feature units.

calibration — A/E, balance, Murphy, rectification

from insurance_monitoring import (
    ae_ratio, ae_ratio_ci,
    check_balance, check_auto_calibration,
    murphy_decomposition,
    rectify_balance, isotonic_recalibrate,
    CalibrationChecker,
)

# A/E ratio with Wilson CI
result = ae_ratio_ci(actual, predicted, exposure=exposure)
print(f"A/E = {result['ae']:.3f}  95% CI [{result['lower']:.3f}, {result['upper']:.3f}]")

# Murphy decomposition — distinguishes miscalibration from discrimination loss
murphy = murphy_decomposition(y=actual, y_hat=predicted, exposure=exposure, distribution="poisson")
print(f"DSC (discrimination score): {murphy.discrimination:.4f}")
print(f"MCB (miscalibration):       {murphy.miscalibration:.4f}")
print(f"Verdict: {murphy.verdict}")   # 'RECALIBRATE' or 'REFIT'

# Full calibration audit in one call
checker = CalibrationChecker(distribution="poisson")
report = checker.check(actual, predicted, exposure=exposure)

discrimination — Gini drift

from insurance_monitoring import (
    gini_coefficient,
    gini_drift_test_onesample,
    GiniDriftBootstrapTest,
)

# One-sample design: tests monitor data against a stored training Gini scalar
# (more natural for deployed model monitoring — reference data may not be available)
result = gini_drift_test_onesample(
    training_gini=0.42,
    monitor_actual=current_claims,
    monitor_predicted=current_predicted,
    monitor_exposure=current_exposure,
)
print(f"Gini change: {result.gini_change:+.3f}  p={result.p_value:.3f}  [{result.significant}]")

# Class-based API with governance plot
bt = GiniDriftBootstrapTest(
    training_gini=0.42,
    monitor_actual=current_claims,
    monitor_predicted=current_predicted,
    monitor_exposure=current_exposure,
    n_bootstrap=500,
)
bt_result = bt.test()
print(bt.summary())      # governance-ready paragraph (method on the test class, not the result)
bt.plot()                # bootstrap histogram with CI shading — IFoA/PRA deliverable

calibration.PITMonitor — anytime-valid calibration monitoring (v0.7.0)

The standard pattern — run a Hosmer-Lemeshow test or check A/E each month — inflates the false-positive rate. After twelve months at α=0.05, the chance of a false alarm exceeds 40% even if the model is perfectly calibrated.

PITMonitor solves this using probability integral transform e-processes (Henzi, Murph, Ziegel 2025, arXiv:2603.13156). The guarantee is: P(ever raise an alarm | model calibrated) ≤ α, for all t, forever.

PITMonitor accepts pre-computed probability integral transforms (PITs) — values in [0, 1] computed from the model's predictive CDF. You compute the PIT for each observation, then pass it to monitor.update():

from insurance_monitoring import PITMonitor
from scipy.stats import poisson

monitor = PITMonitor(alpha=0.05)

# For each new observation: compute PIT from model's predictive CDF, then update
for row in live_data:
    mu = row.exposure * row.lambda_hat          # model's predicted mean
    pit = float(poisson.cdf(row.claims, mu))    # probability integral transform
    alarm = monitor.update(pit)
    if alarm.triggered:
        print(f"Calibration alarm: evidence = {alarm.evidence:.2f}")
        break

sequential — anytime-valid A/B testing (v0.8.0)

Champion/challenger pricing experiments are routinely run with a pre-specified end date. Checking early to stop a bad challenger is statistically invalid under classical hypothesis testing. SequentialTest uses mixture SPRT (Johari et al. 2022) — you can check at every interim update and stop early if the challenger is clearly better or worse, with full type I error control.

from insurance_monitoring import SequentialTest

test = SequentialTest(metric="frequency", alpha=0.05)

# Monthly updates — stop whenever the e-value crosses the threshold
for batch in monthly_batches:
    result = test.update(
        champion_claims=batch.champion_claims,
        champion_exposure=batch.champion_exposure,
        challenger_claims=batch.challenger_claims,
        challenger_exposure=batch.challenger_exposure,
    )
    print(f"e-value: {result.lambda_value:.2f}  stopped: {result.should_stop}")
    if result.should_stop:
        break

drift_attribution — TRIPODD (v0.4.0+)

PSI tells you that driver_age has drifted. TRIPODD tells you whether that driver_age drift explains why the model's discrimination has fallen — accounting for feature interactions. Use InterpretableDriftDetector (v0.7.0) for high-dimensional feature sets or when exposure weighting is required.

from insurance_monitoring import InterpretableDriftDetector

detector = InterpretableDriftDetector(
    error_control="fdr",    # Benjamini-Hochberg for d >= 10 factors
    loss="poisson_deviance", # canonical for frequency models
)
detector.fit_reference(X_ref, y_ref, weights=exposure_ref)
result = detector.test(X_cur, y_cur, weights=exposure_cur)
print(result.significant_features)  # list of features that explain the performance shift

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Regulatory context

PRA SS1/23 (Model Risk Management, March 2023) requires insurers to maintain a model monitoring framework that detects deterioration in model performance. The expectation is documented thresholds, regular testing, and a governance process that escalates to the model risk committee when thresholds are breached. A/E ratio and Gini monitoring are the two metrics most commonly cited in SS1/23 supervisory discussions.

Consumer Duty (PS22/9) requires ongoing monitoring of whether pricing outcomes are fair across customer groups, not just at point of sale. The combination of MonitoringReport and insurance-fairness calibration_by_group() produces a per-protected-group A/E split suitable for Consumer Duty Outcome 4 monitoring.

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Expected performance

On a 50,000-policy synthetic UK motor portfolio:

Task	Time	Notes
PSI on one feature	< 0.1s	Exposure-weighted
CSI across 10 features	< 0.5s	Returns Polars DataFrame
ae_ratio_ci()	< 0.1s	Wilson CI
MonitoringReport (no Murphy)	< 2s	A/E + Gini + CSI
MonitoringReport with Murphy	< 5s	Adds MCB/DSC decomposition
GiniDriftBootstrapTest (n_bootstrap=500)	5–15s	Percentile CI
SequentialTest batch update	< 0.1s	Per monthly update
InterpretableDriftDetector (10 features)	30–90s	FDR-controlled bootstrap

Run the full workflow on Databricks

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Related libraries

Library	What it does
insurance-fairness	Per-protected-group A/E calibration, proxy detection, Consumer Duty audit reports
insurance-causal	Double ML — establishes whether a rating factor causally drives risk or is a proxy
insurance-governance	Model risk committee packs and FCA Consumer Duty documentation
insurance-gam	Interpretable GLM-style models whose factors can be monitored directly

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